A348776 The numbers >= 2 with 3 repeated.

2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset

1,1

Comments

This sequence, 2, 3, 3, 4, 5, 6, 7, ..., gives the stable range of the polynomial rings Z, Z[x_1], Z[x_1, x_2], Z[x_1, x_2, x_3], ...

A note on terminology: "stable range" and "stable rank" are the same thing. In the English-speaking world, people have always used the term "stable range", which was what Bass had invented in the early '60s. When Russian workers wrote on this theme, of course they used a Russian translation of the term "stable range". When the term was translated back into English, it became "stable rank"! - T. Y. Lam, Nov 07 2021

References

T. Y. Lam, Excursions in Ring Theory, in preparation, 2021. See Section 24.

Links

Table of n, a(n) for n=1..83.

F. Grunewald, J. Mennicke, and L. Vaserstein, On the groups SL_2(Z[x]) and SL_2(k[x, y]), Israel J. Math., 86(1-3):157-193, 1994.

Luc Guyot, The stable rank of Z[x] is 3, arXiv:2111.02965 [math.AC], November 2021.

MathOverFlow, Bass' stable range of 𝐙[𝑋]

L. N. Vaseršteĭn and Andrey Aleksandrovich Suslin, Serre's Problem on Projective Modules over Polynomial Rings, and Algebraic K-theory, Mathematics of the USSR-Izvestiya 10.5 (1976): 937 (Russian version).

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = n for n >= 3.

From Chai Wah Wu, Aug 09 2022: (Start)

a(n) = 2*a(n-1) - a(n-2) for n > 4.

G.f.: x*(x^3 - x^2 - x + 2)/(x - 1)^2. (End)

Prog

(Python)
def A348776(n): return n+int(n<3) # Chai Wah Wu, Aug 09 2022

Crossrefs

Sequence in context: A257682 A029928 A101788 * A024698 A011883 A034886

Adjacent sequences: A348773 A348774 A348775 * A348777 A348778 A348779

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 07 2021, following a suggestion from L. Guyot and T. Y. Lam.

Status

approved